Band-geometry criterion for fractional Chern insulator stability

For a partially filled isolated Chern band ($C=1$) of width $W$ with Coulomb scale $U=e^2/(\epsilon a_M)$, Berry curvature $\Omega(k)$ and quantum metric $g(k)$, determine the boundary between the $\nu=1/3$ and $2/3$ fractional Chern insulators and competing charge-density-wave or Fermi-liquid states as a function of $W/U$, the Berry-curvature variance, and the trace-condition violation $T = (1/2 \pi) \int \mathrm{d}^{2}k [\mathrm{tr}\,g(k) - \left| \Omega(k) \right|]$. An answer is a phase diagram validated by exact diagonalization or DMRG across twisted $\mathrm{MoTe_2}$, rhombohedral graphene and model bands.

Systematic DMRG or exact-diagonalization phase diagrams over band families with tunable geometry, checked against experimental FQAH windows.